All the sixdigit numbers that can be formed using the digits

All the six-digit numbers that can be formed using the digits 2,3,5,6,7 and 9 are written. If each digit can be used only once, what is the largest of these numbers that is divisible by 11

Solution

The largest number that can be formed using the given digits without any repitition is 976532

Now, for divisiblity by 11, we have a rule that, the difference in the sum of digits at even places and sum of the digits at odd places of the number should be zero

This is not the case with 976532, because (9+6+3) - (7+5+2) = 4, hence it is not divisible by 11

So, we have to make the sum of numbers at even and odd places equal

Next biggest number would be 976523,

(9+6+2) - (7+5+3) = 2, therefore not divisible by 11

Similarily, other numbers in decreasing order of their magnitude are - 976352, 976325, 976253, 976235, 975632, 975623 etc.

We see that out of these numbers 975623 meets the requisite condition, i.e. (9+5+2) - (7+6+3) = 0

Hence, it is the largest number divisible by 11

Another way to check divisibility is see the difference between the number formed by first three and last three digits, the number obtained should be divisible by 11

for 976532, 976-532 = 444, this is not divisible by 11 hence 976532 is not divisible by 11

and we check the numbers in decreasing order like previously, we will find that for 975623, 975-623 = 352 which is divisible by 11, hence the answer